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924

Appendix HI.

strained by the load, the length and cross section of every ba1
being known, the total work done can be expressed. When thi*
has the lowest possible value, the true stresses in the extra baf^*
will have been found : hence the expression is differentiate^
with regard to each ' redundant ' stress, and equated to zerc?1
The other method, the one which we shall here explain, is
understood by an examination of two cases.

Case I. — Imagine a properly closed structure, without
bars, consisting, say, of four triangles, No load is applied froit1
without, and there is therefore no stress in any member. NoW
let there be two superfluous members, which, bejjig simply fitted
to the framing, are not under stress : neither are the other bar»*»
stressed. Next, screw up the extra bars so as to put a stress
(compressive or tensile) within them, and immediately stressed
are felt upon all the other members. The work done on the
extra bars will be of opposite sign to that done by them indi*
vidually upon the remaining members, for, while one action tends
to compress, the other tends to expand the framing. Also these*
two quantities must be equal, for one is caused by the other.

Let the extra bars be called a and b respectively, and the
remaining bars r, 2, 3, 4, 5 and 6.    Also let

Fa      =  total stress in bar a

>?   >?

„   „   i caused by Fn

j?    n    2        )?        >j     -^b

Then, taking the action of each force Pa and Fb separately,
work done on bar a  =  worlc done on other bars
(minus)                             (plus)

P            F

(i)

(2)

Clearly, then, there would be just as many equations as there are
extra bars.

Case 2 L — Taking the same structure, let external forces be
applied to it.    Firstly, treat the whole figure statically, leaving